Howe duality and the quantum general linear group
Author:
R. B. Zhang
Journal:
Proc. Amer. Math. Soc. 131 (2003), 26812692
MSC (2000):
Primary 17B37, 20G42, 17B10
Published electronically:
December 30, 2002
MathSciNet review:
1974323
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: A Howe duality is established for a pair of quantized enveloping algebras of general linear algebras. It is also shown that this quantum Howe duality implies Jimbo's duality between and the Hecke algebra.
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Howe, R., Perspectives on invariant theory. The Schur Lectures (1992). Eds. I. PiatetskiShapiro and S. Gelbart, BarIlan University, 1995. MR 96e:13006
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Kostant, B., On Macdonald's function formula, the Laplacian and generalized exponents, Adv. Math. 20 (1976) 257285. MR 58:5484
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Kirillov, A. N., Reshetikhin, N., Weyl group and a multiplicative formula for universal matrices. Comm. Math. Phys. 134 (1990) 421431. MR 92c:17023
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Mathas, A., IwahoriHecke algebras and Schur algebras of the symmetric group. Providence, R.I. : American Mathematical Society (1999). MR 2001g:20006
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Montgomery, S., Hopf algebras and their actions on rings. CBMS Regional Conference Series in Math., 82. American Mathematical Society, Providence, RI, 1993. MR 94i:16019
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Parshall, B.; Wang, J. P., Quantum linear groups. Mem. Amer. Math. Soc. 89 (1991), no. 439. MR 91g:16028
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Reshetikhin, N. Yu.; Takhtadzhyan, L. A.; Faddeev, L. D., Quantization of Lie groups and Lie algebras. Algebra i Analiz 1 (1989) 178206. (Russian) MR 90j:17039
 [Ta]
Takeuchi, M., Some topics on . J. Algebra 147 (1992) 379410. MR 93b:17055
 [APW]
 Andersen, H. H.; Polo, P.; Wen, K. X., Representations of quantum algebras. Invent. Math. 104 (1991) 159. MR 92e:17011
 [CP]
 Chari, V.; Pressley, A., A guide to quantum groups. Cambridge University Press, Cambridge, 1994. MR 95j:17010
 [GZ]
 Gover, A. R.; Zhang, R. B., Geometry of quantum homogeneous vector bundles and representation theory of quantum groups. I. Rev. Math. Phys. 11 (1999) 533552. MR 2000j:81108
 [Ho]
 Howe, R., Perspectives on invariant theory. The Schur Lectures (1992). Eds. I. PiatetskiShapiro and S. Gelbart, BarIlan University, 1995. MR 96e:13006
 [Ji]
 Jimbo, M., Quantum matrix related to the generalized Toda system: an algebraic approach. In Field theory, quantum gravity and strings (Meudon/Paris, 1984/1985), 335361, Lecture Notes in Phys., 246, Springer, Berlin (1986). MR 87j:17013
 [Ko]
 Kostant, B., On Macdonald's function formula, the Laplacian and generalized exponents, Adv. Math. 20 (1976) 257285. MR 58:5484
 [KR]
 Kirillov, A. N., Reshetikhin, N., Weyl group and a multiplicative formula for universal matrices. Comm. Math. Phys. 134 (1990) 421431. MR 92c:17023
 [Ma]
 Mathas, A., IwahoriHecke algebras and Schur algebras of the symmetric group. Providence, R.I. : American Mathematical Society (1999). MR 2001g:20006
 [Mon]
 Montgomery, S., Hopf algebras and their actions on rings. CBMS Regional Conference Series in Math., 82. American Mathematical Society, Providence, RI, 1993. MR 94i:16019
 [PW]
 Parshall, B.; Wang, J. P., Quantum linear groups. Mem. Amer. Math. Soc. 89 (1991), no. 439. MR 91g:16028
 [FRT]
 Reshetikhin, N. Yu.; Takhtadzhyan, L. A.; Faddeev, L. D., Quantization of Lie groups and Lie algebras. Algebra i Analiz 1 (1989) 178206. (Russian) MR 90j:17039
 [Ta]
 Takeuchi, M., Some topics on . J. Algebra 147 (1992) 379410. MR 93b:17055
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Additional Information
R. B. Zhang
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, New South Wales 2006, Australia
Email:
rzhang@maths.usyd.edu.au
DOI:
http://dx.doi.org/10.1090/S0002993902068922
PII:
S 00029939(02)068922
Received by editor(s):
June 24, 2001
Received by editor(s) in revised form:
April 7, 2002
Published electronically:
December 30, 2002
Communicated by:
Dan M. Barbasch
Article copyright:
© Copyright 2002
American Mathematical Society
